Extension of a module
Any module containing the given module A as a submodule. Usually one fixes a quotient module X/A, that is, an extension of the module A by the module B is an exact sequence 0 \rightarrow A \rightarrow X \rightarrow B \rightarrow 0 \ .
Such a module X always exists: for example, the direct sum of A and B always forms the split extension; but X need not be uniquely determined by A and B. Both in the theory of modules and in its applications there is a need to describe all different extensions of a module A by a module B. To this end one defines an equivalence relation on the class of all extensions of A by B as well as a binary operation (called Baer multiplication) on the set of equivalence classes, which thus becomes an Abelian group \mathrm{Ext}^1_R(A,B), where R is the ring over which A is a module. This construction can be extended to n-fold extensions of A by B, i.e. to exact sequences of the form 0 \rightarrow A \rightarrow X_{n-1} \rightarrow \cdots \rightarrow X_0 \rightarrow B \rightarrow 0 corresponding to the group \mathrm{Ext}^n_R(A,B). The groups \mathrm{Ext}^n_R(A,B), n=1,2,\ldots, are the derived functors of the functor \mathrm{Hom}_R(A,B), and may be computed using a projective resolution of A or an injective resolution of B. An extension X of A is called essential if S = 0 is the only submodule of X with S \cap A = 0 (that is, A is an essential submodule of X). Every module has a maximal essential extension and this is the minimal injective module containing the given one.
For references see Extension of a group.
Comments
The minimal injective module containing A is called the injective hull or envelope of A. The notion can be defined in any Abelian category, cf. [a1]. The dual notion is that of a projective covering.
References
[a1] | C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) |
Extension of a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_module&oldid=39567